1 Introduction
Nowadays, worldwide spending in digital advertising is skyrocketing, and this growth is primarily driven by ad auctions. These account for almost all market share, since they are at the core of popular advertising platforms, such as, e.g., those by Google, Amazon, and Facebook. According to a recent report by eMarketer [eMarketer], digital ad spending will reach over $490 billion in 2021 and zoom past half a trillion in 2022.
We study signaling in ad auction settings by means of the Bayesian persuasion framework kamenica2011bayesian. Over the last years, this framework has received considerable attention from the computer science community, due to its applicability to many realworld scenarios, such as, e.g., online advertising bro2012send; emek2014signaling; badanidiyuru2018targeting, voting alonso2016persuading; cheng2015mixture; castiglioni2019persuading; semipublic, traffic routing vasserman2015implementing; bhaskar2016hardness; castiglioni2020signaling, recommendation systems mansour2016bayesian, security rabinovich2015information; xu2016signaling, and product marketing babichenko2017algorithmic; candogan2019persuasion.
In a standard ad auction, the advertisers (also called bidders) compete for displaying their ads on a limited number of slots, and each bidder has their own private valuation representing how much they value a click on their ad. In this work, we study Bayesian ad auctions, which are characterized by the fact that bidders’ valuations depend on a random, unknown state of nature. The auction mechanism has complete knowledge of the actual state of nature, and it can send signals to bidders so as to disclose information about the state and increase revenue. In particular, the auction mechanism commits to a signaling scheme, which is defined as a randomized mapping from states of nature to signals being sent to the bidders. Our model fits many realworld applications that are not captured by classical ad auctions. For instance, a state of nature may collectively encode some features of the user visualizing the ads—such as, e.g., age, gender, or geographical region—that are known to the mechanism only, since the latter has access to data sources unaccessible to the bidders.
We study the problem of computing a revenuemaximizing signaling scheme for the mechanism. In particular, in this paper we focus on public signaling, in which the mechanism can only send a single signal that is observed by all the bidders. Moreover, we restrict our attention to VCG mechanisms, which are widely used in practice and have the appealing property of inducing bidders to truthfully report their valuations. While the signaling problem studied in this paper has already been addressed in the easier setting of secondprice auctions badanidiyuru2018targeting, to the best of our knowledge, our work is the first to explore algorithmic signaling in general ad auctions with more than one slot.
1.1 Original Contributions
We start our analysis with a negative result, showing that, in general, the revenuemaximizing problem with public signaling does not admit a PTAS unless , even when bidders’ valuations are known to the mechanism. Thus, in the rest of the paper, we address settings in which we can prove that such a negative result can be circumvented.
First, we show that, in the known valuations setting, the problem admits a polynomialtime algorithm when either the number of slots or the number of states is fixed. The proposed algorithms work by solving suitablydefined linear programs (LPs) of polynomial size, thanks to the crucial property that, when either or is fixed, there always exists an optimal signaling scheme using a polynomial number of different signals. Moreover, we also study special instances in which the bidders are single minded, but and can be arbitrary. In this case, each bidder positively values a click on their ad only when the actual state of nature is a specific (single) state, and all the bidders interested in the same state value a click on their ad for the same amount. By exploiting a particular combinatorial structure of the set of bidders’ posterior distributions induced by signaling schemes, we are able to provide an FPTAS in such setting. The algorithm works by applying the ellipsoid method in a nontrivial way, with only access to an approximate polynomialtime separation oracle. The latter is implemented by a rather involved dynamic programming algorithm, which works thanks to the particular structure of the set of bidders’ posteriors.
Then, we switch the attention to the random valuations setting, where bidders’ valuations are unknown to the mechanism, but randomly drawn according to some probability distribution. In this case, we first provide some preliminary results that establish useful connections between the optimal value of the revenuemaximizing problem and that of optimal signaling schemes restricted to suitablydefined finite sets of posterior distributions. These sets are defined so that the expected revenue of the mechanism is “stable”, meaning that it does not decrease too much when restricting signaling schemes to use posteriors in such sets. In particular, for our results we use sets of uniform posteriors, for suitable values of . As a preliminary step, we also show that it is possible to compute an approximatelyoptimal signaling scheme having only access to a finite number of samples from the distribution of bidders’ valuations. In conclusion, all the preliminary results described so far allow us to prove that, in the random valuations setting, the problem admits an FPTAS, a PTAS, and a QPTAS, when, respectively, is fixed, is fixed, and bidders’ valuations are bounded away from zero.^{1}^{1}1All the proofs are in the Supplementary Material.
1.2 Related Works
To the best of our knowledge, the algorithmic study of signaling in auctions is limited to the secondprice auction, which can be seen as a special ad auction with a single slot.
emek2014signaling [emek2014signaling] study secondprice auctions in the known valuations setting. They provide an LP to compute an optimal public signaling scheme. Moreover, they show that it is hard to compute an optimal signaling scheme in the random valuations setting. In our work, we generalize their positive result, in order to provide our polynomialtime algorithm working when the number of slots is fixed.
cheng2015mixture [cheng2015mixture] complement the hardness result of emek2014signaling by providing a PTAS for the random valuations setting. This result cannot be extended to ad auctions, as we show in our first negative result. However, we provide two generalizations of the result by cheng2015mixture [cheng2015mixture]: we provide a PTAS for the random valuations setting with a fixed number of slots , and a QPTAS when the bidder’s valuations are bounded away from zero.
Finally, badanidiyuru2018targeting [badanidiyuru2018targeting] study algorithms whose running time does not depend on the number of states of nature. Moreover, they initiate the study of private signaling schemes, showing that, in secondprice auctions, private signaling introduces nontrivial equilibrium selection problems.
2 Preliminaries
In a standard ad auction (see also the book by nisan2001algorithmic [nisan2001algorithmic] for more details), there is a set of advertisers (or bidders) who compete for displaying their ads on a set of slots, with . Each bidder is characterized by a private valuation , which represents how much they value a click on their ad. Moreover, each slot is associated with a click through rate parameter , which is the probability with which the slot is clicked by a user.^{2}^{2}2In this work, for the ease of presentation, we assume that the click through rate only depends on the slot and not on the ad being displayed. In general, each slot may have its own prominence value—the probability with which a user observes it—and each bidder may have their own ad quality—the probability with which their ad is clicked once observed—, so that the click through rate is defined as the product of these two quantities. All the results in this paper can be easily extended to such general model. W.l.o.g., we assume that the sots are ordered so that . The auction goes on as follows: first, each bidder separately reports a bid to the auction mechanism; then, based on the bids, the latter allocates an ad to each slot and defines how much each bidder has to pay the mechanism for a click on their ad. We focus on truthful mechanisms, and the VCG mechanism in particular (see the book by mas1995microeconomic [mas1995microeconomic] for a complete description of the mechanism). In truthful mechanisms, allocation and payments are defined so that it is a dominant strategy for each bidder to report their true valuation to the mechanism, namely for every . In particular, the allocation implemented by the VCG mechanism orderly assigns the first bidders in decreasing value of to the first slots (those with the highest click through rates). At the same time, assuming w.l.o.g. that bidder is assigned to slot , the mechanism defines an expected payment for each bidder , where, for the ease of notation, we let . The payment is zero for all the other bidders. In practice, each bidder has to pay whenever a user clicks on their ad, so that their utility is in expectation over the clicks. The expected utility of all the other bidders is zero.
We study Bayesian ad auctions, which are characterized by a set of states of nature. Each bidder
has a valuation vector
, with being bidder ’s valuation in state , and all such vectors are arranged in a matrix of bidders’ valuations , whose entries are defined as for all and . We model signaling by means of the Bayesian persuasion framework kamenica2011bayesian. We consider the case in which the auction mechanism knows the state of nature and acts as a sender by issuing signals to the bidders (the receivers), so as to partially disclose information about the state and increase revenue. As customary in the literature, we assume that the state is drawn from a common prior distribution , with denoting the probability of state .^{3}^{3}3Given a finite set , we denote with the ()dimensional simplex defined over the elements of . The mechanism publicly commits to a signaling scheme , which is a randomized mapping from states of nature to signals for the bidders. We focus on the case of public signaling in which all the bidders receive the same signal from the auction mechanism. Formally, a signaling scheme is a function , where is a set of available signals. For the ease of notation, we let be the probability of sending signal when the state is .A Bayesian ad auction goes on as follows (see Figure 1 for a picture): (i) the auction mechanism commits to a signaling scheme , and the bidders observe it; (ii) the mechanism gets to know the state of nature and draws signal ; and (iv) the bidders observe the signal and rationally update their prior belief over states according to Bayes rule. After observing signal , all the bidders infer a posterior distribution over states (also called posterior
for short) such that the posterior probability of state
is(1) 
Finally, each bidder truthfully reports to the mechanism their expected valuation given the posterior , namely , and the mechanism allocates slots and defines payments as in a standard ad auction.
Representing Signaling Schemes.
It is oftentimes useful to represent signaling schemes as convex combinations of the posteriors they can induce dughmi2014hardness; cheng2015mixture. Formally, a signaling scheme induces a probability distribution over posteriors in , with denoting the probability of posterior , defined as
Indeed, we can directly reason about distributions over rather than about signaling schemes, provided that they are consistent with the prior. By letting be the support of , this requires that
(2) 
In the rest of the paper, we will use the term signaling scheme to refer to a consistent distribution over .
Computational Problems.
We focus on the problem of computing an optimal signaling scheme, i.e., one maximizing the revenue of the mechanism. We study two settings:

the known valuations (KV) setting in which the matrix of bidders’ valuations is known to the mechanism; and

the random valuations (RV) setting in which the matrix of bidders’ valuations is unknown, but randomly drawn according to a probability distribution .
As it is customary in the literature (see, e.g., badanidiyuru2018targeting), in the RV setting we assume that algorithms have access to a blackbox oracle returning i.i.d. samples drawn from (rather than actually knowing such distribution). We denote by the expected revenue of the mechanism when the bidders’ valuations are given by and the posterior induced by the mechanism is . Formally, given that bidders truthfully report their expected valuations and assuming w.l.o.g. that bidder is assigned by the mechanism to slot , we can write . Then, given a signaling scheme , the expected revenue of the mechanism is . When the valuations are unknown, we let and define the expected revenue analogously. Notice that, given a distribution of valuations (or, in the KV setting, a matrix of bidders’ valuations ) and a finite set of posteriors, it is possible to formulate the problem of computing an optimal signaling scheme as an LP, as follows:^{4}^{4}4LP 3 is written for the RV setting, its analogous for the KV setting can be obtained by substituting with .
(3a)  
(3b) 
In the following, we let be the optimal value of LP 3, while we denote with the optimal expected revenue of the mechanism over all the possible signaling schemes .^{5}^{5}5The dependence of and from either or is omitted, as it will be clear from context.
3 A General Inapproximability Result
We start our analysis with the following negative result:
Theorem 1.
The problem of computing an optimal signaling scheme does not admit a PTAS unless , even when it is restricted to the KV setting.
Theorem 1 is proved by a reduction from the VERTEX COVER problem in cubic graphs APXAlimonti.
4 KV Setting: Parametrized Complexity
In this section, we study the parametrized complexity of the problem of computing an optimal signaling scheme, showing that it admits a polynomialtime algorithm when either the number of slots or the number of states of nature is fixed.
In the following, we let be the set of all the the possible permutations of bidders taken from , with denoting a tuple made by bidders , in that order. We also let be the (possibly empty) polytope of posteriors in which the expected valuations of bidders in are ordered (from the highest to the lowest) according to ; formally, it holds . Notice that, given a permutation of bidders, the expected revenue of the mechanism in any posterior is , since the bidders truthfully report their expected valuations to the mechanism, and, thus, the latter allocates slots to bidders in according to their order in the permutation. Thus, for any fixed with , the term is linear in over .
4.1 Fixing the Number of Slots
In this case, the problem can be solved in polynomial time by formulating it as an LP, thanks to the following lemma:
Lemma 1.
There always exists an optimal signaling scheme such that for every .
Intuitively, the lemma follows from the fact that, given any signaling scheme and two posteriors such that for some , it is always possible to define a new signaling scheme that replaces and with a suitablydefined convex combination of them, without decreasing the expected revenue (since it is linear over ).
By Lemma 1, we can rewrite the revenue maximization problem as subject to constraints ensuring that each belongs to (for ) and that is a consistent probability distribution over such posteriors (see Equation (2)). This problem can be formulated as an LP by introducing a variable for each and , encoding the products that define the expected revenue. Overall, the resulting LP (see LP 8 in the Supplementary Material) has a number of variables and constraints that is , which, after fixing , is polynomial in the size of the input.^{6}^{6}6We remark that LP 8 in the Supplementary Material is a generalization of the LP presented by emek2014signaling [emek2014signaling] for the easier case of second price auctions. Thus, we conclude that:
Theorem 2.
In the KV setting, if the number of slots is fixed, then an optimal signaling scheme can be computed in polynomial time.
4.2 Fixing the Number of States
Our polynomialtime algorithm exploits the fact that an optimal signaling scheme can be computed by restricting the attention to distributions supported on a finite set of posteriors whose cardinality is polynomial in all the parameters, except from . In particular, it is sufficient to focus on the set , where denotes the set of vertices of the polytope given as input. Formally:
Lemma 2.
It holds that .
The lemma follows from the fact that, given any signaling scheme and posterior such that for some , by Carathéodory’s theorem it is always possible (since is a polytope) to decompose into a convex combination of the vertices of , obtaining a new signaling scheme that provides the mechanism with an expected revenue at least as large as that of (since is linear over ). By observing that , it is easy to show that an optimal signaling scheme can be computed by means of LP 3 instantiated for the set , which has a number of variables and constraints that is polynomial once is fixed. This proves the following:
Theorem 3.
In the KV setting, if the number of states is fixed, then an optimal signaling scheme can be computed in polynomial time.
5 KV Setting: SingleMinded Bidders
In this section, we focus on particular Bayesian ad auctions where the bidders are single minded. Intuitively, in our setting, by single mindedness we mean that each bidder is interested in displaying their ad only when the realized state of nature is a specific (single) state, and that all the bidders interested in the same state value a click on their ad for the same amount. We introduce the following formal definition:
Definition 1 (Singleminded bidders).
In a Bayesian ad auction, we say that bidders are single minded if there exist and for all such that:

and for all ;

for every and , it holds and for all .
Notice that, given a posterior induced by the mechanism, all the bidders belonging to the same set have the same expected valuation, namely for all and . As a result, given that bidders truthfully report their expected valuations, the mechanism will always receive at most different bids, one per set .
The last observation implies that, given , in order to unequivocally define an allocation of bidders to slots (and, thus, also define the expected payments) it is sufficient to know the relative ordering of the (at most) different expected valuations associated to sets . This allows us to tackle the problem with an approach analogous to the one of Section 4, with the only difference that, in this case, we will reason about permutations of the groups of bidders , rather than about permutations of all the individual bidders.
In the following, we let be the set of all the permutations of the sates of nature , while we let be an ordered tuple made by states , where . Moreover, is the polytope of posteriors in which the expected valuations associated to sets are ordered according to .
The first preliminary result that we need in order to derive our approximation algorithm is a characterization of the vertices of the sets for , as follows.
Lemma 3.
Given and , it holds that if and only if there exists such that:

; and

.
Intuitively, Lemma 3 states that the vertices of a set are all the posteriors such that, for some , only the first states according to the ordering defined by are assigned a positive probability, while all the remaining states have zero probability. Moreover, the positive probabilities of the posterior are defined so that all the bidders belonging to the first sets , according to the ordering defined by , are the same. Notice that, in the special case in which all the values are equal to one, the vertices of all the sets are all the uniform probability distributions over subsets of states of nature, for any .
By letting , since the term is linear in over for every permutation , we can conclude that (the proof is analogous to that of Lemma 2). Thus, Lemma 3 allows us to find an optimal signaling scheme by solving LP 3 for the set and the matrix of bidders’ valuations . However, notice that, since the size of is exponential in , the resulting LP has exponentiallymany variables. Nevertheless, since the LP has polynomiallymany constraints, we can still solve it in polynomial time by applying the ellipsoid algorithm to its dual, provided that a polynomialtime separation oracle is available.
In order to design a polynomialtime separation oracle, we apply the procedure described above to a relaxed version of LP 3, whose optimal value is sufficiently “close” to that of the original LP. Given , the relaxed LP reads as follows:
(4a)  
(4b) 
The dual problem of LP 4 reads as follows:
(5a)  
(5b)  
(5c) 
where for are dual variables associated to Constraints (3b), while is a dual variable for . Notice that, by relaxing the LP, in the dual LP 5 we get the additional Constraint (5c) and that for all . This is crucial to design a polynomialtime separation oracle.
The separation problem associated to Problem 5 reads as:
Definition 2 (Separation problem).
Given values for the dual variables for all , compute:
(6) 
The following Lemma 4 shows that Problem 6 can be solved optimally up to any given additive loss , by means of a dynamic programming algorithm that runs in time polynomial in the size of the input, in , and in . Formally:
Lemma 4.
Given , there exists an algorithm that finds an additive approximation to Problem 6, in time polynomial in the size of the input, in , and in .
The crucial observation that allows us to solve Problem 6 by means of dynamic programming is that, in any posterior , bidders’ expected valuations are either a positive, bidderindependent value or zero (see Lemma 3). This allows us to build a discretized range of possible bidders’ valuation values, so that, for each discretized value, we can compute an optimal posterior inducing that value by adding states of nature incrementally in a dynamic programming fashion.
Since the algorithm in Lemma 4 only returns an approximate solution to Problem 6, we need to carefully apply the ellipsoid algorithm to solve LP 5, so that it correctly works even with an approximated oracle. Some nontrivial duality arguments allow us to prove that, indeed, this can be achieved by only incurring in a small additive loss on the quality of the returned solution, and without degrading the running time of the algorithm. Overall, this allows us to conclude that:
Theorem 4.
In the KV setting, if the bidders are single minded, then the problem of computing an optimal signaling scheme admits an (additive) FPTAS.
6 RV Setting
In this setting, as stated in Section 2, we assume that the auction mechanism has access to the distribution of bidders’ valuations only through a blackbox sampling oracle. In the following, given i.i.d. samples of matrices of bidders’ valuations, namely , we let be their empirical distribution, which is such that for all .
In this section, we first study the parametrized complexity of the problem of computing an optimal signaling scheme in general auctions (Section 6.1), and, then, we address special auction settings in which the bidders’ valuations are bounded away from zero, namely for all and , for some threshold . In the latter case, we show that the problem admits a QPTAS and the result is tight (Section 6.2).
Before stating our main results (Theorems 5, 6, 7, and 8), we introduce some preliminary useful lemmas. The first one (Lemma 5) works under the true distribution of bidders’ valuations , and it establishes a connection between the optimal expected revenue () and the optimal value of LP 3 for suitablydefined finite sets of posteriors (). In particular, we look at sets for which the function is “stable” according to the following definition:^{7}^{7}7Notions of stability analogous to that in Definition 3 have already been used in the literature; see, e.g., cheng2015mixture.
Definition 3 (stability).
Given and a finite set , we say that is stable for if, for every , there exists a distribution such that:
(7) 
For any finite set such that is stable for , starting from an optimal signaling scheme one can recover an optimal solution to LP 3, only incurring in “small” multiplicative and additive losses in the expected revenue, respectively of and . This can be accomplished by decomposing each posterior into and, then, putting such distributions together. These observations allow us to prove the following lemma:
Lemma 5.
Given and such that is stable for , it holds .
The second lemma (Lemma 6) deals with the approximation error introduced by using an empirical distribution of bidders’ valuations , rather than the actual distribution . Given a finite set of posteriors, let be an optimal solution to LP 3 for distribution and set . Moreover, let be the average expected revenue of signaling schemes under the true distribution of valuations , where the expectation is with respect to the sampling procedure that determines . Then, a concentration argument proves the following:
Lemma 6.
Given , let be finite and , .
Finally, the last lemma (Lemma 7) exploits Lemma 5 to provide two useful bounds on the value of , where (for a given ) is the finite set of all the uniform posteriors, according to the following definition:
Definition 4 (uniform posterior).
Given , a posterior is uniform if each is a multiple of .
Notice that the set has size . The two points in the following lemma are readily proved by applying Lemma 5, after noticing that the sets in the statement are such that the function is stable for them, with suitable values of and . Formally:
Lemma 7.
Given and , it holds:

;

if, for some , it is the case that for all and , then .
6.1 Parametrized Complexity
First, we study the computational complexity of the problem of computing an optimal signaling scheme when the number of states is fixed. We provide an (additive) FPTAS that works by performing the following two steps: (i) it collects a suitable number of matrices of bidders’ valuations, by invoking the sampling oracle; and (ii) it solves LP 3 for the resulting empirical distribution and a suitablydefined set of uniform posteriors. In particular, given a desired (additive) error , the algorithm works on the set for and its approximation guarantees rely on the following Lemma 8, proved again by means of Lemma 5.
Lemma 8.
For and , .
Thanks to Lemmas 6 and 8 (the former applied for suitable values ), we can prove that the procedure described in steps (i) and (ii) above gives a signaling scheme achieving an expected revenue at most a function of lower than , provided that the number of samples is defined as in Lemma 8. Moreover, let us notice that, since , if is fixed, then the overall procedure runs in time polynomial in the input size and in . Thus, we can conclude that:
Theorem 5.
In the RV setting, if the number of states is fixed, then the problem of computing an optimal signaling scheme admits and (additive) FPTAS.
Next, we switch the attention to the case in which the number of slots is fixed. We provide an (additive) PTAS that works as the FPTAS in Theorem 5, but whose approximation guarantees follow from Lemma 6 and point (i) in Lemma 7 (rather than Lemma 8). Thus, the only difference with respect to the previous case is that the algorithm works on the set of uniform posteriors for defined as in Lemma 7. As a result, since and depends on a parameter that is related to the quality of the obtained approximation, the algorithm is only a PTAS rather than an FPTAS. Formally, we can prove the following:
Theorem 6.
In the RV setting, if the number of slots is fixed, then the problem of computing an optimal signaling scheme admits and (additive) PTAS.
6.2 Valuations Bounded Away From Zero
We conclude the section by studying the case in which the bidders’ valuations are bounded away from zero. This case is dealt with an algorithm identical to the one in Theorem 6, but carrying on the approximation analysis by using Lemma 6 and point (ii) in Lemma 7 (rater than point (i)). Thus, since the value of in Lemma 7 is related to the quality of the approximation thorough a parameter and also depends logarithmically on the number of slots , we obtain:
Theorem 7.
In the RV setting, if for all and for some , then the problem of computing an optimal signaling scheme admits a (multiplicative) QPTAS.
The following theorem shows that the result is tight.
Theorem 8.
Assuming the ETH, there exists a constant such that finding a signaling scheme that provides an expected revenue at least of requires time, where is the size of the problem instance. This holds even when for all and .^{8}^{8}8The notation hides polylogarithmic factors.